Complete (d+1)-uniform hypergraph is generically circumsphere rigid

Prove that, for every dimension d and number of points n satisfying binomial(n, d+1) ≥ d n − binomial(d+1, 2), the complete (d+1)-uniform hypergraph is generically circumsphere rigid in dimension d; that is, for a generic ordered point cloud P in R^d, the circumsphere framework (H, P) with H equal to the complete (d+1)-uniform hypergraph is locally identifiable up to Euclidean isometry from the collection of circumradii associated to its hyperedges.

Background

To analyze identifiability for Čech persistence, the authors introduce a new rigidity theory for circumsphere frameworks, where constraints are given by circumradii of point subsets (hyperedges). They show circumsphere rigidity is a generic property and formulate combinatorial questions paralleling classical rigidity theory for graphs.

The conjecture proposes a sharp combinatorial threshold for generic circumsphere rigidity of the complete (d+1)-uniform hypergraph, mirroring classical results in Euclidean rigidity. The authors report computational verification via Jacobian rank tests for d ≤ 5, but a general proof is lacking.